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F[x_] := ((x - a)^2 + b^2)*((x - c)^2 + d^2)
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F1[x_] := ((x)^2 - 2 a x + (a^2 + b^2))*((x)^2 - 2 c x + (c^2 + d^2))£¬
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FullSimplify[Integrate[1/F[x], x]]=(-d (-b^2 + (a - c)^2 + d^2) ArcTan[(a - x)/b] +
b (-((a - c)^2 + (b - d) (b + d)) ArcTan[(c - x)/
      d] + (a - c) d (-Log[b^2 + (a - x)^2] +
       Log[d^2 + (c - x)^2])))/(b d ((b^2 + (a - c)^2)^2 +
   2 (a - b - c) (a + b - c) d^2 + d^4))
ºÍFullSimplify[Integrate[1/F1[x], x]]=((b^2 + 2 a c) d ArcTan[(a - x)/b] +
   b (2 a c + d^2) ArcTan[(c - x)/d] + a^2 d ArcTan[(-a + x)/b] +
   c^2 d ArcTan[(-a + x)/b] + d^3 ArcTan[(-a + x)/b] +
   a^2 b ArcTan[(-c + x)/d] + b^3 ArcTan[(-c + x)/d] +
   b c^2 ArcTan[(-c + x)/d] - a b d Log[b^2 + (a - x)^2] +
   b c d Log[b^2 + (a - x)^2] +
   b (a - c) d Log[d^2 + (c - x)^2])/(b d ((b^2 + (a - c)^2)^2 +
     2 (a - b - c) (a + b - c) d^2 + d^4))
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